Discretization schemes that preserve non-trivial symmetry groups of partial differential equations have received increasing attention over the past 20 years. These investigations parallel ongoing efforts to construct conservative discretizations and belong to the field of geometric numerical integration. It has been found that preserving symmetry groups for evolution equations is in general not possible on stationary discretization meshes. The problem of invariant discretization is therefore intimately linked to the problem of grid adaptation. In this talk we will introduce different strategies that allow constructing discretization schemes with symmetry properties on r-adaptive meshes. The proposed strategies rely on the use of invariant evolution–projection techniques and on invariant meshless discretizations. We will illustrate the different approaches by carrying out invariant numerical integrations for several evolution equations with relevance in geophysical fluid dynamics.